Lunar phase is very important to astronomers, though perhaps not for the reason you think. Most astronomers regard the moon as a nuisance, whose light interferes with observation of deep-sky objects and distant galaxies.
For deep-sky astronomers, the most prized time to observe is during the "dark time" when the moon is between 3rd quarter and 1st quarter, a two-week period centered on the time of new moon. The least desirable time to observe is during "bright time," when the moon is between 1st quarter and 3rd quarter.
Lunar astronomers prefer to observe when the moon is near 1st quarter or 3rd quarter. Around these times, the lighting conditions of the moon are most favorable for showing fine details on the lunar surface. Full moon is actually the worst time to observe because all the surface details of the moon are "washed out."
At least, lunar astronomers and deep-sky astronomers find something to agree on when the moon is full. It's just not a very attractive time to observe much of anything!